Does Peat Creep? Canvas Version

Peatlands are dynamic, deformable porous media storing over 30% of global soil carbon. Researchers in the MATHPEAT Network are currently investigating the evidence for peat creep, how it should be represented mathematically, and what its consequences may be for peatland age-depth structure and long-term carbon storage.

The model below is deliberately simple. It is not a full 3D landscape-scale creep model: creep mostly redistributes peat locally. Here we ask what a single representative column would look like under different vertical creep profiles and depth-scaling assumptions if unresolved downslope or marginal creep eventually exports material from that column.

This comparison version uses the same equations and controls as the main interactive post, but the plots are drawn directly with the browser’s HTML <canvas> element rather than Chart.js.

Simple Column Model & Creep Formulations

Let $M(a)$ be cumulative peat dry mass per unit area at cohort age $a$, measured in $\mathrm{kg\,m^{-2}}$. The estimated peat depth is

\[H(a) = \frac{M(a)}{\rho_b},\]

where fixed dry bulk density is $\rho_b = 100\,\mathrm{kg\,m^{-3}}$.

Without creep, the classic accumulation-decay balance (Clymo 1984) is

\[\frac{dM}{da} = A - \alpha M, \qquad M(0)=0,\]

where $A$ is the accumulation flux (default $60\,\mathrm{g\,m^{-2}\,yr^{-1}}$) and $\alpha$ is the first-order decay rate (default $10^{-4}\,\mathrm{yr^{-1}}$).

Two Independent Creep Choices

The creep model separates two assumptions that are often mixed together:

  1. Vertical velocity profile: both profile choices restrict lateral creep to an active upper layer of thickness $H_{\text{active}}$ ($H_{\text{plug}}$ or $H_{\text{shear}}$):
    • Uniform / plug-like profile: peat in the active layer moves downslope at a uniform velocity. Below this depth, the peat is stable (no creep).
    • Linear shear-zone profile: speed is maximum at the surface and decreases linearly to zero at depth $H_{\text{shear}}$. Below this depth, the peat is stable.
  2. Depth scaling of measured speed: the user sets the present-day surface creep speed measured at the final simulated depth. The model then reconstructs what that speed would have been when the peat column was shallower using one of four simple hypotheses: constant, proportional to $H$, proportional to $H^2$, or proportional to $1/H$.

Creep Velocity Profiles: Plug vs. Shear Flow

Uniform Plug Flow Active Layer (Creeping) Stable Layer (Stationary) H_active u = u_surf Linear Shear Flow Active Layer (Creeping) Stable Layer (Stationary) H_active u decreases to 0

The Reverse-Engineering Challenge (Equifinality)

In field peatland science, scientists extract a physical peat core, measure its depth $H$, and perform Radiocarbon ($^{14}\text{C}$) dating to get age $a$ at depth $H$.

The Challenge: A given observed age-depth profile could be fitted by TWO completely different mechanisms:

  1. Classic Clymo Model (Decay Only): Fitted with a lower surface accumulation rate $A_{\text{Clymo}}$ or higher decay rate $\alpha_{\text{Clymo}}$.
  2. Peat Creep Model (Decay + Export): Driven by a higher true surface accumulation rate $A_{\text{Creep}}$, standard decay $\alpha_{\text{Creep}}$, and active lateral creep $u_{\text{meas}}$.

If creep is ignored, fitting an observed profile with only Clymo’s equation will underestimate true carbon accumulation rates ($A$) and misattribute lateral creep loss to decomposition!

Interactive Age-Depth & Cumulative Mass Model

Clymo / Yu Model (Decay Only) Teal Line
Present-day surface peat input flux
60 g/m²/yr
First-order catotelm decay rate
1.0e-4 /yr
Accumulation decay rate ($A(a) = A_0 e^{-\beta a}$)
0.0e+0 /yr
Creep Model (Decay + Export) Dashed Red Line
Choose where in the peat column lateral motion is active
How today's measured surface speed is reconstructed for shallower past peat
True surface input flux
60 g/m²/yr
True catotelm decay rate
1.0e-4 /yr
Present-day surface speed at final simulated depth
0.50 cm/yr
Depth limit for lateral motion
1.0 m
Clymo / Yu Depth (10,000 yrs)
-
With Creep Depth (10,000 yrs)
-
Depth Discrepancy
-
Derived Coeff
-

Estimated Depth Profile ($H$, meters)

Cumulative Dry Mass Profile ($M$, kg/m²)

Integrated Lateral Volume Flux ($Q$, m²/yr)

🔍 Step-by-Step Equifinality Challenge (Try It!)

Peatland science and inverse modeling are plagued by unidentifiability. Can you distinguish a creeping bog from climate change? Try this:

  1. Set up a creeping bog: In the red panel (Creep Model), increase true surface accumulation $A_{\text{Creep}}$ to 90 g/m²/yr and measured creep speed $u_{\text{meas}}$ to 2.00 cm/yr. Notice how the dashed red line (Creep model) thins out compared to the baseline.
  2. Scenario A: Ignored Creep (Clymo fit): Keep $\beta = 0$ on the left panel. To match the final depth of the creeping bog, you are forced to drag $A_0$ down to ~52 g/m²/yr. If you ignore creep, your model severely underestimates historical carbon sequestration!
  3. Scenario B: Hiding Creep in Climate Change (Yu fit): Keep the creep profile active. Now, set $A_0$ to 90 g/m²/yr (matching the true creep input) on the left panel, and adjust the temporal change slider $\beta$ to -5.0e-5 /yr. Notice how the solid teal line and the dashed red line match almost perfectly!
    The mathematical catch: A creeping bog behaves identically to a bog where accumulation increased exponentially over time. Without independent mechanical measurements, creep is completely hidden in the climate reconstruction!

Mathematical, Physical & Peatland Science Foundations

1. Peatland Science & Eulerian vs. Lagrangian Views

Standard 1D peat accumulation models (such as Clymo 1984) split the peat column into an upper oxic layer (acrotelm) and a lower water-saturated anoxic layer (catotelm). Organic matter enters the catotelm at surface accumulation flux $A$ ($\text{g m}^{-2}\text{yr}^{-1}$) and undergoes slow anaerobic decay at a first-order rate $\alpha$ ($\text{yr}^{-1}$).

Under steady-state conditions over 10,000 years:

2. Temporal Variations: The Yu et al. (2003) & Belyea & Malmer (2004) Model

Clymo’s model assumes a constant peat addition rate $A$ through time. To allow for climate-driven or developmental variations, the model by Yu et al. (2003) and Belyea & Malmer (2004) permits the initial accumulation rate of cohorts to vary exponentially back through calendar time. Expressed in terms of cohort age $a$ today, the remaining mass of a cohort of age $a$ today is $m(a) = A_0 e^{-(\alpha - \beta) a}$.

Integrating this from age $0$ to $a$ today yields the cumulative peat mass profile:

\[M(a) = \frac{A_0}{\alpha - \beta} \left( 1 - e^{-(\alpha - \beta) a} \right), \qquad \text{for } \alpha \neq \beta\]

and $M(a) = A_0 a$ when $\alpha = \beta$.

Here, $A_0$ is the present-day surface accumulation flux, $\alpha$ is the decomposition rate, and $\beta$ controls the temporal accumulation change. Note that this Lagrangian age-depth profile within a single core today is mathematically identical to Clymo’s constant-accumulation equation but with a shifted effective decay rate, $\alpha_{\text{eff}} = \alpha - \beta$. This is physically distinct from the calendar-time growth history of the total bog height over time, which is $H(t) = \frac{p}{\alpha - \beta}(e^{-\beta t} - e^{-\alpha t})$.

A positive $\beta$ represents a peatland where initial accumulation was higher in the past (decreasing towards the present, resulting in a thicker bottom). A negative $\beta$ represents a peatland where initial accumulation was lower in the past (increasing towards the present, resulting in a thinner bottom).

3. Mathematical Derivation of the Creep Closure

In a 2D/3D continuum, mass conservation is $\frac{\partial M}{\partial t} + \nabla \cdot \left(\mathbf{u} M\right) = A - \alpha M$. In this one-column demo, unresolved lateral divergence is approximated by an export term

\[\frac{dM}{da} = A - \alpha M - \frac{u_{\text{eff}}(H)}{L}M,\]

where $L=100\text{ m}$ is a representative export length. The user-controlled speed $u_{\text{meas}}$ is interpreted as the present-day surface creep speed measured when the simulated core has its final depth $H_{\text{final}}$.

Vertical Profile Choices

Both profile choices assume that lateral creep is restricted to an active upper layer of thickness $H_{\text{active}}$ (denoted $H_{\text{plug}}$ or $H_{\text{shear}}$):

Integrated Lateral Flux

Integrating the lateral velocity profile $u(z)$ over the depth of the peat column gives the integrated lateral volume flux $Q$ (in $\mathrm{m^2\,yr^{-1}}$):

\[Q = \int_0^H u(z) \, dz\]

We can analyze this integral in two distinct ways:

  1. Spatial Cumulative Flux (Present Day):
    • Mathematics: This is an indefinite integral over depth $z$ evaluated at the final present time ($t = T_{\text{final}}$): \(Q_{\text{spatial}}(z) = \int_0^z u(T_{\text{final}}, s) \, ds \qquad \text{for } z \in [0, H_{\text{final}}]\) where $z = H(a)$ is the depth of the cohort of age $a$ today.
    • Physics: It answers: “In the modern peat column today, how much lateral creep flux is carried by the upper layers down to depth $z$?” Because the surface ($z = 0$, cohort age 0) has a thickness of 0, this cumulative value starts at $0$ today and increases to the total column flux at the bottom of the core (age 10,000).
  2. Historical Total Flux (Over Time):
    • Mathematics: This is a definite integral over the entire peat thickness $H(t)$ at varying historical times $t \in [0, T_{\text{final}}]$: \(Q_{\text{historical}}(t) = \int_0^{H(t)} u(t, z) \, dz\) where $t = T_{\text{final}} - a$ is the historical time when the oldest cohort was $a$ years old.
    • Physics: It answers: “At any historical time $t$ in the bog’s history, what was the total lateral creep flux integrated across the entire profile?” Because the bog is mature today (present, age 0), this total flux is at its maximum today. It only drops to $0$ at the initial start of the bog 10,000 years ago (age 10,000).

For the two profile types under an active layer of depth $H_{\text{active}}$ (where $u_{\text{surf}}$ is the surface speed):

This lateral volume flux can be converted to a lateral dry mass flux (in $\mathrm{kg\,m^{-1}\,yr^{-1}}$) by multiplying by the constant bulk density: $F_{\text{lateral}} = Q \cdot \rho_b$.

Depth-Scaling Choices for the Measured Surface Speed

The depth-scaling selector asks how the present-day measured speed should be extrapolated backward when the peat column was thinner:

These are not competing claims about the true law; they are deliberately simple closures for exploring how different physical hypotheses alter the inferred age-depth profile.

4. The Equifinality & Inverse Problem Challenge

When analyzing an observed radiocarbon-dated core from a field site, scientists face an inverse problem (fitting parameters from measured age-depth points).

If lateral creep is active in the bog but omitted from the inversion model:

Hiding Creep in the Climate Inversion (Yu/Belyea Model)

If scientists use the more flexible Yu / Belyea model to fit the core (instead of Clymo’s constant-$A$ model), they introduce the temporal parameter $\beta$. Because lateral creep exports peat and thins the profile at depth, the effects of creep can be completely absorbed into the inferred temporal history.

Sampling Bias in Core Selection

In field campaigns, researchers typically sample cores from the deepest part of the peatland (e.g., the plateau of a raised bog). This introduces a systematic sampling bias: by choosing the deepest and thickest plateaus, scientists preferentially sample exactly the locations where the peat column is oldest, yet where lateral creep stress and peat loss over millennia are also most active.

References